1250 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 5, SEPTEMBER 2006

A Hopfield Neural Network forImage Change Detection

Gonzalo Pajares, Member, IEEE

Abstract—This paper outlines an optimization relaxation ap-proach based on the analog Hopfield neural network (HNN) forsolving the image change detection problem between two images.A difference image is obtained by subtracting pixel by pixel bothimages. The network topology is built so that each pixel in thedifference image is a node in the network. Each node is charac-terized by its state, which determines if a pixel has changed. Anenergy function is derived, so that the network converges to stablestates. The analog Hopfield’s model allows each node to take onanalog state values. Unlike most widely used approaches, wherebinary labels (changed/unchanged) are assigned to each pixel, theanalog property provides the strength of the change. The maincontribution of this paper is reflected in the customization of theanalog Hopfield neural network to derive an automatic imagechange detection approach. When a pixel is being processed, someexisting image change detection procedures consider only inter-pixel relations on its neighborhood. The main drawback of suchapproaches is the labeling of this pixel as changed or unchangedaccording to the information supplied by its neighbors, whereits own information is ignored. The Hopfield model overcomesthis drawback and for each pixel allows a tradeoff between theinfluence of its neighborhood and its own criterion. This is mappedunder the energy function to be minimized. The performance ofthe proposed method is illustrated by comparative analysis againstsome existing image change detection methods.

Index Terms—Change detection, difference images, energy min-imization, Hopfield neural network (HNN).

I. INTRODUCTION

Amajor portion of the research efforts of the computer visioncommunity has been directed towards the study of auto-

matic image change detection methods [1] due to a large numberof applications in diverse disciplines. Important applications ofchange detection include video surveillance [2]–[7], analysisof multitemporal remote sensing images [8]–[10], tracking sys-tems of moving objects [11], medical diagnosis [12], or driverassistance systems [13].

Given a set of images of the same scene taken at several dif-ferent times, the goal is to identify the set of pixels that aresignificantly different between the last image and a previousreference image. Changed pixels may result from a combina-tion of underlying factors, including appearance or disappear-ance of objects, motion of objects relative to the background,

Manuscript received March 7, 2005; revised December 19, 2005. This workwas supported in part by the Fundación General UCM under Grant 143/2004 andby the Departamento de Observacón de la Tierra, Teledetección y Aeronomía,Instituto Nacional de Técnica Aeroespacial.

The author is with the Departamento de Sistemas Informáticos y Progra-mación, Facultad de Informática, Universidad Complutense, Madrid 28040,Spain (e-mail: pajares@dacya.ucm.es).

Digital Object Identifier 10.1109/TNN.2006.875978

shape change of objects or environment modifications (build-ings, fires), etc., [1]. Although in some scenarios a sequence ofmany images is available, our approach works only with twoimages, including images of different sequences. So techniquesbased on background modeling are out of the scope of this paper.The reader is referred to [1]–[7] and related works for this issue.

Before the image detection method is applied, we assume thatthe images have been geometrically registered. In order to avoidstrong intensity variations, we perform, if required, radiometricadjustment through homomorphic filtering according to the re-sults obtained in the work of Pajares et al. [14]. The homomor-phic scheme used is that defined in [15].

We propose a new approach based on the analog Hopfieldneural network (HNN) paradigm for solving the image changedetection problem between two images. A difference image isobtained by subtracting pixel by pixel both images. The net-work topology is built so that each pixel in the difference imageis a node in the network. Each node is characterized by its state,which determines if a pixel has changed. An energy function isderived, so that the network converges to stable states. The HNNmodel allows each node to take on analog state values, i.e., todetermine the strength of the change. In [8] and [16], an energyfunction is minimized taking into account only interpixel rela-tions between a pixel and its neighbors. This implies that thispixel is labeled as changed or unchanged according to the infor-mation supplied by its neighbors and that its own informationis ignored. Our HNN model overcomes this drawback and foreach pixel allows one to achieve a tradeoff between the influ-ence of its neighborhood and its own criterion. This is mappedunder the energy function to be minimized.

This paper is organized as follows. Section II contains a re-view of image change detection techniques. Based on the anal-ysis of such techniques, the motivation and contribution of theproposed method is explained. In Section III, the customizedHNN process is described where the energy function to be mini-mized is derived. In Section IV, we summarize the image changedetection procedure. The performance of the method is illus-trated in Section V, where a comparative analysis against otherexisting image change detection strategies is carried out. Finally,in Section VI, there is a discussion of some related topics.

II. TECHNIQUES IN IMAGE CHANGE DETECTION

A. Image Change Detection: Review

There are four major types of change detection schemes thatapply in detecting the difference between two images andof the same scene taken at different times.

1) Temporal Difference Models: A difference imageis computed at each pixel location by subtracting the

1045-9227/$20.00 © 2006 IEEE

PAJARES: HOPFIELD NEURAL NETWORK FOR IMAGE CHANGE DETECTION 1251

corresponding intensity values of the incoming images, i.e.,. The changed image is

generated according to the following decision rule:

if changeotherwise no change

(1)

The threshold is chosen empirically. This method is verysensitive to changes in the illumination.

Rosin and Ionnidis [4] evaluate quantitatively some clas-sical automatic thresholding algorithms for choosing . Manyhistogram-based thresholding approaches regard finding athreshold in the histogram for binarizing the image differenceas approximating the histogram with two distribution func-tions. This implies that two major peaks and a valley must beidentified; this sometimes involves an intrinsic difficulty dueto histogram fluctuations. In [3], a technique is proposed toavoid such fluctuations based on a cumulative histogram; theproblem is reduced to find a change-point where the trend ofcumulative occurrence rates rigidly changes. In the work ofLu and Suganthan [6], an accumulation algorithm for videodetection is proposed. It is based on small differences betweenconsecutive frames and accumulates them; when the accumu-lation difference exceeds a threshold, a change is declared.

In [5], a modified temporal difference method is proposedfor processing video sequences. Initially, a clean backgroundtemplate image with zero value for each pixel is setfirst. Then the image difference is obtained for con-secutive frames and . After the image differ-ence is obtained, subtracting from the previous tem-plate image yields a difference image

where the pixel of is definedto be a changed pixel if ; otherwise, it belongs tothe background. Finally, assign to for the next frame toproceed with change detection.

A closely related approach to simple differencing is changevector analysis [9], often used for multispectral images. A fea-ture vector is generated for each pixel in the image consideringseveral spectral channels. The modulus of the difference be-tween feature vectors at each pixel location gives the changes.

2) Significance and Hypothesis Tests Models: The decisionrule in some change detection algorithms is cast as a statisticalhypothesis test. The decision as to whether or not a change hasoccurred at a given pixel location corresponds to choosingone of the two competing hypotheses: The null hypothesisor the alternative hypothesis , corresponding to no-changeand change decisions, respectively. The image pair and isviewed as a random vector.

In [16], both hypotheses are characterized by modeling theobservations of the difference image assuming the gray-level differences to obey zero-mean Gaussian distributions withvariances and for and , respectively. The goal ofthis approach is to estimate a changed binary mask given

. As a special case of a Bayesian estimate, is estimated bymaximizing the a posteriori probability . Under theBayesian framework and after an amount of manipulations, the

likelihood and the a priori probabilities are combined under thefollowing decision rule:

and (2)

where is a global threshold, is a neighborhood around thepixel location , and and are the a prioriprobabilities for labeling the pixel as changed and un-changed. They are computed taking into account the relationsbetween the labeled pixels in .

Bruzzone and Fernandez [8] propose an image change detec-tion technique that estimates the parameters of the mixture dis-tribution consisting of all pixels in the difference image.The mixture distribution can be written as

(3)

Under this assumption, the probability density functions, and the a priori probabilities

and are estimated by using the expectationmaximization (EM) algorithm [8], [17], which is a generalapproach to maximum-likelihood estimation. It is assumedthat both and can be modeledby Gaussian distributions. So, the parameters to be estimatedare the means , and variances , , respectively. Theprocess is iterated until convergence and the initial values ofthe estimates are determined by simple differencing. Once theabove parameters are estimated, the method considers spatialinformation. It is assumed that a pixel belonging to a class islikely to be surrounded by pixels belonging to the same class.Under this assumption and making use of the Markov randomfields framework, a conditional distribution of a pixel label ismodeled. This distribution is based on the Gibbs energy func-tion that is to be minimized. This energy function is the sum oftwo terms; the first describes the interpixel class dependencetaking into account that the pixels in the same neighborhoodwith similar labels decrease the energy; the second computesthe logarithmic values of or fora pixel and its neighbors. All values are summed and the energydecreases as the logarithmic values increase. The algorithm isiterated until convergence.

3) Vector and Shading Models: This model is also knownas linear dependence detector. It is a method proposed by Du-rucam and Ebrahimi [18] and based on linear algebra. Each pixelat location of images can be represented by one vector of

neighboring intensity pixels, . Therefore,whether or not the same pixel is changed in different frames,with , can be validated if both representativevectors are linearly dependent or independent. Let , be twolinearly dependent vectors with no components zero; then theratio of their components is constant, i.e.,

1252 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 5, SEPTEMBER 2006

. In an eight-connected neighbor-hood, this is expressed as

constant. It follows that the variance must be zero

(4)

where is the number of members in the neighborhood of. The decision is made as

change (linearly independent)no change (linearly dependent)

(5)

The shading model proposed in the work of Skifstad and Jain[19] is closely related to the vector model. The shading modelsgenerally compare the ratio of image intensities to a thresholddetermined empirically. Assuming shading models, the inten-sity image at a pixel can be modeled as the product oftwo components: The illumination from the light sourcein the scene and the reflectance of the object surfaceto which belongs, i.e., . Sincethe reflectance component depends only on the intrinsic prop-erties of the object surface imaged at , given two images

and in the absence of a change, we can write. Hence, the ratio of image intensities can be modeled

as follows:

(6)

Hence, if the illuminations and in eachimage are approximated as constant within the neighborhood

, the ratio of intensities remains constantunder the null hypothesis . This justifies a null hypothesis thatassumes linear dependence between vectors of correspondingpixels intensities. Skifstad and Jain [19] implement the shadingapproach through (4) and (5).

In [11], a shading approach is also proposed. It is based onthe assumption that only intensities of the scene can be mea-sured and the reflectance function is not available. Also, takinginto account the effects of additive noise, a probability densityfunction is derived. Given the neighborhood of size andintensity images as frames , the - or -directional thorder circular moments are

(7)

where and . Given twoframes and , compute

(8)

where .With a fixed threshold , a significant change occurs in if

for some

(9)

4) Clustering Models: Carlotto [20] computes image sta-tistics over clusters and assumes that background pixel valueswithin clusters can be modeled as Gaussian distributions aboutmean values that vary cluster-to-cluster, and that anomalies havepixel values that deviate significantly from the distribution ofthe cluster. This deviation is measured through the Mahalanobisdistance [17]. The clustering is carried out by vector quantiza-tion (VQ). In one dimension, the VQ algorithm assigns imagepixel values to one of levels. The levels

are chosen so as to minimize the mean square quanti-zation error (MSQE)

MSQE (10)

where is the probability distribution over intensity valuesin the image.

Assume the reference image has been divided intointensity levels so that the VQ algorithm produces a set ofclusters . Each multiband data pixel in theimage is classified as belonging to one of the clusters,

. Each cluster is modeled as a Gaussian distribu-tion with mean and covariance matrix . Over the set ofpixel locations in a given reference cluster , a different set ofpixels values are observed in the test image obtaining thecluster with mean and covariance matrix . If thereare changes affecting this cluster, the test image mean and co-variance may be different from the reference image. Based onthis hypothesis, it is conjectured that changed pixels in a clusterwill have higher Mahalanobis distance values than unchangedpixels.

B. Motivational Research

The previously mentioned approaches are currently popularand useful methods for image change detection. In spite of their

PAJARES: HOPFIELD NEURAL NETWORK FOR IMAGE CHANGE DETECTION 1253

relative simplicity and widespread use, some of the aforemen-tioned change-detection methods are nonautomatic and requireempirical experiments [4], [16], [18]. Indeed, in classical tech-niques, the analysis of the difference image is performed bythresholding such image according to empirical strategies ormanual trial-and-error procedures. Some techniques have triedto avoid this problem by using automatic thresholding, but thisrequires that the changes be significant [3], [4]. Another im-portant drawback is based on the assumption that only a fewchanges occurred in the study area between the two images.This implies that techniques searching for minima/maxima onthe histogram of the image difference fail [3]. Also, clusteringmethods require that the above assumption be true [20]. In videosurveillance, a common approach uses both background estima-tion and foreground extraction [5], [21], but this is not possiblein applications in which a ground truth is not available, i.e., re-mote sensing images acquired at different times. Another in-herent difficulty when evaluating difference images is posed bythe presence of noise, which gives raise false changes [16].

Some approaches have tried to solve the aforementionedproblems. Of particular interest are the methods in [8], [9], and[16]. They try to minimize the thresholding, noise, or missingground truth problems by implementing an automatic imagechange detection technique that considers spatial-contextualinformation. Such approaches are based on the hypothesis thata changed/unchanged pixel is surrounded by pixels of identicalnature. According to the description of significance and hy-pothesis tests models, the spatial-contextual information comesfrom two sources [see (2) and (3)]: Gaussian distributions forthe difference image and changed/unchanged labels.

C. Contribution of This Paper

In [8] and [16], the labeled information in the neighborhoodof a pixel is mapped as an energy function taking into accountthe interlabel relations. So, under similar labels, the energy de-creases. Also in [8], the energy decreases if the pixel under eval-uation and its neighbors have high probability values computedfrom Gaussian distributions. This is called spatial-contextual in-formation.

According to the results reported in [8], [9], and [16], it isclear that an efficient use of the spatial contextual informationimproves the change detection performance. So, we propose amethod that makes use of this information under its two cat-egories, renamed as a) data consistency coming from the dif-ference image and b) contextual consistency extracted from thepixels that have been already labeled as changed or unchanged.The main drawback of using only consistencies is that the la-beling of each pixel depends on the labels of its neighbors only,where its own contribution is ignored. This effect may lead toincorrect labels, particularly in the borders of changed areas.Our HNN approach allows one to achieve a tradeoff betweenthe influence of the neighborhood and the own contribution andtries to solve the problem by modeling the own contributionas self-data information. Data and contextual consistencies andself-data information are mapped under the form of an energyfunction which is to be minimized by optimization. We also in-troduce an improvement in the mapping of the data consistency.Indeed, instead of summing the statistics of the gray levels in the

difference image as in [8], we consider interdata relations, sothat similar data values (high or low) decrease the energy. Ad-ditionally, the analog HNN model is suitable for computing thestrength of the change for each pixel based on the state value inthe corresponding node. This represents an improvement withrespect some existing strategies [3]–[5], [8], [16], where onlybinary labels (changed/unchanged) are assigned to each pixel.

III. IMAGE CHANGE DETECTION BY THE HNN

A. A Review on the HNN

The HNN paradigm initially proposed by Hopfield [22] hasbeen widely used for solving optimization problems. This im-plies fixing two characteristics [23]: Its activation dynamics andan associated energy function which decreases as the networkevolves.

The HNN is a recurrent network containing feedback pathsfrom the outputs of the nodes back into their inputs so thatthe response of such a network is dynamic. This means thatafter applying a new input, the output is calculated and fedback to modify the input. The output is then recalculated, andthe process is repeated again and again. Successive iterationsproduce smaller and smaller output changes, until eventuallythe outputs become constant, i.e., at this moment the networkachieves an acceptable stability.

The connection weights between the nodes in the networkmay be considered to form a matrix . Although some recentstudies carried out by Lee and Chuang [24] in HNNs have beenaddressed for solving the problem of optimal asymmetric as-sociative memories, we have found acceptable the classical ap-proach studied in [25], where it is shown that a recurrent net-work is stable if the matrix is symmetrical with zeros on its di-agonal, that is, if for all and and for allneurons [25], [26].

To illustrate the Hopfield networks in more detail, considerthe special case of a Hopfield network with a symmetric matrix.The input to the node comes from two sources: Externalinputs and inputs from the other nodes. The total input tonode is then

(11)

where the value represents the output of the th node;is the weight of the connection between nodes and ; andrepresents an external input bias value which is used to set thegeneral level of excitability of the network through constant bi-ases. There are two kinds of Hopfield networks [23], [27]: 1)Analog, in which the states of the neurons are allowed to varycontinuously in an interval, such as [ 1, 1]; and 2) discrete,in which these states are restricted to the binary values 1 and

1. The drawback of these binary networks is that they oscil-late between different binary states and settle down into one ofmany locally stable states. Hopfield has shown that analog net-works perform better since they have the ability to smooth thesurface of the energy function which prevents the system frombeing stuck in minor local minima [22], [28].

For analog Hopfield networks, the total input into a node isconverted into an output value by a sigmoid monotonic activa-

1254 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 5, SEPTEMBER 2006

tion function instead of the thresholding operation for discreteHopfield networks [26]. The dynamic of a node is defined by

where (12)

is the sigmoid activation function and is a time con-stant that can be set to one for simplicity [27], [29]. We havechosen the sigmoid activation function to be the hyperbolic tan-gent function [27], . This function is differ-entiable, smooth, and monotonic, i.e., contributes to the networkstability [26]. A detailed discussion about the settings of the timestep and gain can be found in [23]. As increases, theprobability that the energy falls into a local minimum also in-creases. According to some experiments carried out by Joya etal. [23] where this parameter has been set to values in the range1 to 10 , the best performance is achieved with the minimumvalue (i.e., 10 ). Hence we have fixed it to 10 , which is anorder of magnitude less than the experiment in [23]. The way toavoid that a continuous network cannot find a solution due to theexistence of local minimum and makes the network converge upto a solution state is to decrease along the simulation, theoret-ically until . This strategy reminds a simulated annealingprocess starting from high enough . Then the network evolvesuntil a stable state (which is not a solution) is reached, then isdecreased and the network evolves again up to a new stable state,and so on; the process ends when becomes zero and at thismoment, the stable state reached should be a global minimum.According to the results reported in [30] and [31], we have testedthe following scheduling strategy where

is the iteration number. We have computed as follows [32].1) We select eight pairs of images, where the nodes have been

initialized; now we compute the initial energy.2) We choose an initial that permits about 80% of all tran-

sitions to be accepted (i.e., transitions that decrease theenergy function), and this value is changed until this per-centage is achieved.

3) We compute the transitions and look for a valuefor for which , afterrejecting the higher order terms of the Taylor expansion ofthe exponential, , where is the mean value.

In our experiments, we have obtained , giving. In the work of Starink and Backer [30], a simulated

annealing scheduling is used with , i.e., with the sameorder of magnitude. Taking into account that ,

, and considering , we obtain , i.e.,. In our image change detection approach, we have carried out

different experiments by applying the above scheduling and alsoassuming fixed gain to be three without apparent improvementin the final results. Hence we set the gain to three during the fullprocess. The quantity describing the state of the network, calledenergy, is defined as follows:

(13)

According to the results reported in [23], the integral term in(13) is bounded by when is 1 or 1 and is nullwhen is zero. In our experiments, we have verified that thisterm does not contribute to the network stability and only theenergy is increased in a very little quantity with respect to theother two terms in (13); hence for simplicity we have removedit from (13).

The continuous Hopfield model described by the system ofnonlinear first-order differential equation (12) represents a tra-jectory in phase space, which seeks out the minima of the energyfunction in (13).

B. Image Change Detection by HNN

Under the analog HNN paradigm, the problem of imagechange detection is to label each pixel of the incoming imagesas changed or unchanged and the degree of change. Withsuch purpose, we consider the output image as a network ofnodes where each node is associated to a pixel location inthe difference image, i.e., the number of nodes is exactly thenumber of pixels of the incoming images. Also, each nodeis characterized by its state value , ranging in [ 1, 1].The network state is characterized by the states of the nodes.After the optimization process, when the network stability isreached, the nodes have achieved its final state value. Thisfinal value will indicate unchanged ( 1) or maximum change( 1); intermediate values give the strength of the change. Theoptimization process minimizes the energy in (13) startingfrom an initial network state. According to the terminologyin Section II-C, the interconnection weights in (13) arecomputed by applying data and contextual consistencies andthe external inputs through the self-data information.

1) Network Initialization: The network initialization is car-ried out by exploiting the characteristics of the difference image.We use the initialization strategy, described in [8], as follows.From the histogram of the difference image (see Sec-tion II-A), we compute two thresholds and as

and , where is the middle valueof , i.e., , and ,set to 0.5 in this process. Now, given a pixel location inthe difference image, its associated node in the network is ini-tialized as follows:

ififor randomly otherwise .

(14)

This initialization can be exploited to analyze the perfor-mance of HNN against other strategies.

2) Energy Function for Image Change Detection: An im-portant issue addressed in neural computation is how sensoryelements in a scene perceive the objects, i.e., how the sceneanalysis problem is addressed. To deal with real-world scenes,some criterion for grouping elements in the scene is required.In the work of Wang [33], a list of major grouping principles isexhaustively studied. In our image change detection approach,we apply the following three principles: proximity, changed/un-changed pixels that lie close in space tend to group; similarity,

PAJARES: HOPFIELD NEURAL NETWORK FOR IMAGE CHANGE DETECTION 1255

changed pixels with similar values tend to group; and connect-edness, changed/unchanged pixels that lie inside the same con-nected region tend to group.

Based on the above principles, now the problem is to buildan energy function where data and contextual consistencies andalso self-data information can be mapped.

a) Consistency from the data: According to the descrip-tion in Section II-A (hypothesis tests models) and the work ofBruzzone and Fernandez [8], the data information is mappedthrough the a posteriori probabilities that given a pixel value

in the difference image is associated to hypothesis, i.e., we compute . This is car-

ried out by applying the Bayes rule

(15)

where the density functions and the a prioriprobabilities are estimated through the EM algorithm.The mixture density distribution is computedthrough (3). The initialization required by the EM algorithmis based on the method described in Section III-B1, underthe assumption that is with hypothesis or if

or , respectively. As we do nothave prior information, the a priori probabilities are initializedto 0.5.

Now, each pixel in the difference image shouldbe associated to the hypothesis that maximizes the posteriorconditional probability, for example

(16)

From (16), we build a data map with the same size as thedifference image and identical locations that those of thepixels in the image difference and nodes in the network. Eachnode in the location is loaded with the data information

according to the criterion in (16) as follows:

(17)

Now, the goal is to map the data consistency between nodesand into the consistency coefficient . Given the node , weconsider its -connected neighborhood under the groupingcriterion established by the proximity and connectedness prin-ciples according to [33]; could be 4, 8, 24, 48, or any othervalue taking into account only horizontal, vertical or diagonaldirections. A typical value is eight, corresponding to a centralpixel and its eight neighbors.

For each node , only consistencies can be established be-tween nodes , where and ; otherwise, if ,it is assumed that there is not consistency between nodes and

. This is justified under the hypothesis that only local relationscan be established between changed/unchanged pixels. Indeed,given a changed pixel, probably its neighbors should be also

changed pixels and vice versa for unchanged pixels. Two nodes, where are said consistent if they have similar data

information. Otherwise they should be inconsistent.The data consistency between the nodes , is mapped into

the coefficient as follows:

.(18)

ranges in [ 1,+ ]. The contribution so made may bepositive (excitatory synapse) or negative (inhibitory synapse).Hence, a positive data consistency will contribute towards thenetwork stability. This is the interdata relation contributionstated in Section II-C.

b) Consistency from the contextual information: In someexisting works [8], [9], [16], [31], the interpixel dependence isdescribed by defining a kind of consistency which is achievedunder the consideration of contextual information. We make useof this concept and map it so that it can be embedded also underour HNN approach. Given the node at the pixel locationwith state value and a set of nodes with state values

, a measurement of contextual consistency between the nodeand its neighbors can be expressed as

(19)

This term represents an interstate relation. As , rangein [ 1, 1], given , the term will be maximum whenthe values are close to . Indeed, assuming that under theeight-neighbourhood and take simultaneously values of

1 or 1, , i.e., reaches its maximum value. On thecontrary, if and all or and all

, , i.e., its minimum value. It is worthnoting that (19) can be regarded as an implementation of theGibbs potential in a neighborhood under the Markov randomfields framework [8], [16], [31].

Once data and contextual consistencies are specified, wesearch for an energy function such that the energy is low whenboth consistencies are high and vice versa. This energy isexpressed as

(20)

where is a positive constant to be defined later, sgn is thesignum function, and is the number of negative values in theset , i.e., given ,

card .Table I shows the behavior of the energy term against

data and contextual consistencies. As one can see, the energydecreases as the data and the states are both simultaneously con-sistent (rows 1 and 4 in the left part of the Table I); otherwise

1256 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 5, SEPTEMBER 2006

TABLE IBEHAVIOR OF THE ENERGY TERM AGAINST

DATA AND CONTEXTUAL CONSISTENCIES

under any inconsistency the energy increases. We have consid-ered that data inconsistencies have higher priority than contex-tual ones; so under this criterion if then the energyincreases.

c) Self-data information: We have analyzed the interre-lations between nodes in a given neighborhood, based on dataand contextual consistencies. This implies that the state for eachnode evolves according to the information provided by the ma-jority in the neighborhood, ignoring its own information. Thismay lead to an incorrect state for the node under consideration.To overcome this drawback, we assume that each node mustcontribute to the evolution of its own state through the self-datainformation (Section II-C). The self-data information is mod-eled as a kind of self-consistency based on the hypothesis that anode with high probability of change must be labeled as changedand vice versa. This implies that a node with high/low data in-formation must have a high/low state value simultaneously.Under this assumption, the self-consistency is mapped as an en-ergy function as follows:

(21)

The constant B is a positive number to be defined later. So,given the node with low/high and values simultane-ously imply that is minimum, as expected.

3) Derivation of the Connection Weights and the External In-puts for the HNN: Assuming data and contextual consistencies(20) and self-data information (21), we derive the energy func-tion of the (22), which is to be minimized by optimization underthe HNN framework

(22)

By comparison of (13) and (22) without the integral term in(13), it is easy to derive the connection weights and the externalinput bias as follows:

(23)

According to the discussion in Section III-A, to ensure theconvergence to stable state [25], symmetrical interconnectionweights and no self-feedback are required, i.e., we see that bysetting , both conditions can easily be derived from(23). The energy in (23) represents a tradeoff between the data

and contextual information coming from the spatial environ-ment surrounding the node (pixel) and also its self-data infor-mation. The constants A and B could be fixed so that they tunethe influence of each term in (23). We have carried out severalexperiments verifying that in our approach the above setting isappropriated.

Equation (12) describes the time evolution of the network.The total input to the neuron is computed by solving (12)with the Runge–Kutta method. Finally, the state is also com-puted according to (12). As we can see, the energy in (22) isobtained by considering the state values and a kind of attractive-ness derived from both data and contextual consistencies. Thederivation of an energy function with attractiveness betweenfixed points has been well addressed in the work of Müezzinogluet al. [34] for discrete Hopfield memories preserving symmet-rical weights and without self-feedback. Hence, we can assumethat under the attractiveness of data and contextual consisten-cies, our analog Hopfield approach performs appropriately.

Fig. 1 shows a pedagogical example about the behavior ofthe energy function, based on the contribution of the and

terms. Assume a changed pixel located at the central posi-tion in the window; the states and data consistencies are shownseparately. The difference image ranges from 0 to 255, i.e., 0 isunchanged, 255 changed, and . This pixel belongs to aborder changed region defined by the four locations with valuesset to 255.

Based on the central pixel , we compute andaccording to (20) and (21), respectively. So, the total

energy value for the central pixel is . We can see thatthe addition of to leads to a total energy value whichis less than , i.e., the network gains stability thanks to thecontribution of . This is also applicable if the central pixelwas unchanged.

IV. SUMMARY OF THE IMAGE CHANGE DETECTION PROCEDURE

After mapping the energy function onto the HNN, the imagechange detection process is achieved by letting the networkevolve so that it reaches a stable state, i.e., when no changeoccurs in the states of its nodes during the updating procedure.The whole image change detection procedure can be summa-rized as follows.

1) Initialization: Create a node for each pixel locationfrom the difference image; (iteration number); loadeach node with the state value as given by (14); compute

, through (23); (a constant to accelerate theconvergence); (maximum number of iterationsallowed); set the constant values as follows: ;

; . Define as the number of nodes thatchange their state values at each iteration.

2) HNN process: Set and ; for each nodecompute using the Runge–Kutta method and update

, both according to (12) and ifthen ; when all nodes have been updated, if

and then go to step 2) (new iteration),else stop.

3) Outputs: updated for each node (changed/unchangedpixel locations).

PAJARES: HOPFIELD NEURAL NETWORK FOR IMAGE CHANGE DETECTION 1257

Fig. 1. Pedagogical example about the behavior of the energy terms.

TABLE IIDATA SETS PROPERTIES AND DESCRIPTION

V. DESCRIPTIONS OF DATA SETS AND EXPERIMENTS

A. Description of the Data Sets

In order to assess the validity and performance of the pro-posed method, for the analysis of the image change detectiontask, we considered four different data sets: Real video se-quences of outdoor and indoor environments and real andsynthetic remote sensing images. Table II summarizes the dif-ferent data sets used and their properties. Note that the number

of nodes in the network is exactly the size of each frame.A comparative analysis is carried out against some existingstrategies as explained below.

This analysis requires a ground truth for each pair of imagesanalyzed, useful to assess change-detection errors and successes.So, we prepare a map of changed unchanged areas as follows.Rosin and Ioannidis [4] evaluate several classical global imagethresholding approaches for image change detection. Based onthat study, the best performance is achieved with the method de-

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Fig. 2. Outdoor environment: (a) and (b) two images of the same sequence, (c) ground truth map, (d) network initialization, and (e) changes detected with theHNN approach.

Fig. 3. Indoor environment: (a) and (b) two images of the same sequence (intervals A and B, respectively), (c) ground truth map, and (d) changes detected withthe HNN approach (homomorphic filtering is applied).

scribed in the work of Kapur et al. [35], which uses the entropyof the histogram. In [3], also a new technique based on cumu-lative histograms is used with acceptable performance. By ap-plying both methods we obtain two binary images. They are com-bined by using the logical operator “or” giving a resulting image,which is manually refined. The manual refinement is carriedout through a visual inspection and removing spurious changedareas or pixels or adding missing changed regions. After thisprocess, the ground truth map is obtained.

Figs. 2–4 show representative pairs of images belonging tothe type of data described in the Table II. In Fig. 2, (a) and (b)show frames and (the subscript indicates the number offrames in the sequence) of the same outdoor sequence; (c) shows

the ground truth map; (d) shows the results of the network ini-tialization; and (e) shows the results obtained by the HNN algo-rithm after four iterations. The maximum degree of change cor-responds to black points and unchanged pixels to white areas. Itis worth noting how the HNN approach evolves from the initial-ization image to the final result removing an important numberof false changed and unchanged areas. Fig. 3(a) and (b) showstwo images of the same indoor sequence but taken during timingintervals A and B, respectively, i.e., under different illumina-tion conditions (homomorphic filtering is applied). Fig. 3(c) and(d) shows the ground truth map and the results obtained by theHNN method, respectively; the changes are represented with theabove black and white criterion.

PAJARES: HOPFIELD NEURAL NETWORK FOR IMAGE CHANGE DETECTION 1259

Fig. 4. (a) and (b) Urban area acquired by the IKONOS satellite at different times. (c) and (d) Synthetic images obtained from (b) of mean zero and variance 5.0and “salt and pepper” of density 0.2, respectively. (e) Ground truth map. (f) Changes obtained for the HNN method between (a) and (c).

TABLE IIIWINDOW SIZE AND THRESHOLD VALUES FOR THE CHANGE DETECTION METHODS

Fig. 4(a) and (b) shows two images of the same urban areaacquired during different days; Fig. 4(c) and (d) shows noisyimages corrupted from (b) with zero-mean Gaussian noise ofvariance 0.5 and “salt and pepper” noise of density 0.2 respec-tively; Fig. 4(e) shows ground truth map; and Fig. 4(f) showschanges obtained by the proposed HNN approach between im-ages (a) and (c). A common practice in image change detectionapproaches is the corruption of images by adding some type ofnoise, i.e., the generation of synthetic corrupted data sets. Thisallows assessing more accurately the robustness of the proposedtechniques against different levels of noise [5], [8], [11], [30].The zero-mean Gaussian noise with different variances is themost used. We have also added “salt and pepper” as this kind ofnoise allows us to verify the behavior of our approach against adispersion of black and white isolated points.

B. Description of the Experiments

Different experiments have been carried out to asses the va-lidity and robustness of the proposed HNN approach. The ef-fectiveness of our HNN method is verified against the followingsix strategies, described in Section II: MTD [5], LIU [11], MAP[16], SKI [19], CAR [20], and BRU (spatial-contextual informa-tion) [8]; Table III shows the thresholds and window sizes usedfor each method. The window size is the neighborhood in HNN,BRU, and MAP required for mapping the contextual informa-tion. We have tested window sizes of 5 5 and 7 7 without ac-curacy improvements. The results obtained for each method arecompared against the ground truth based on the magnitudes [4]

TP true positives, i.e., number of change pixels correctlydetected;

FP false positives, i.e., number of no-change pixelsincorrectly labeled as change;

TN true negatives, i.e., number of no-change pixelscorrectly detected;

FN false negatives, i.e., number of change pixelsincorrectly labelled as no-change.

From these quantities the following two measures areused [4]: The percentage of correct classification computedas PCC (TP TN)/(TP TN)(TP FP TN FN) and theYule coefficient, as in the work of Sneath and Sokal [36], is(TP/(TP FP)) (TN/(TN FN))-1 . The most commonly used

for assessing classifiers is PCC but in sequences where theamount of change is small compared to the overall image ittends to give misleading estimates. This is minimized by usingthe Yule coefficient. We used both coefficients.

HNN and BRU are iterative automatic change detectionmethods; the threshold row specifies the threshold valuesused for each method according to the discussion in the cor-responding references where such methods are described. Inour HNN approach, the threshold is the parameter valueaccording to the discussion in Section III-B1. Note that HNNuses the same initialization process as BRU. In [8], it is re-ported that the initialization threshold is a noncritical parameterbecause the iterative processes assume initial possible errors.Hence, this assertion is also applicable for our HNN strategy.

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TABLE IVSET OF EXPERIMENTS AND PAIRS OF IMAGES SELECTED

Fig. 5. Averaged energy values for the set of eight experiments, according to the three types of images analyzed: (a) and (b) video sequences for outdoor andindoor environments, respectively, and (c) remote sensing images.

This implies that any threshold value could be used for bothHNN and BRU, i.e., the automatic condition for both methodsis preserved.

Table IV describes the set of experiments carried out from thedata sets shown in the Table II. It indicates the number of pairsand the characteristics of the images selected . MTD re-quires a sequence of images; hence it is not tested for experi-ments E2, E6, E7, and E8. For the remainder of the experiments,the sequence of frames used in MTD is defined by the framescaptured between and , where is the reference frame and

is the last frame.

C. Experimental Results

The comparative performance is analyzed in terms of the cor-rect classification, computation time, and number of iterations.Previously, we show some results related intrinsically with theHNN behavior.

Fig. 5 shows the energy behavior of each experiment againstthe number of iterations necessary to reach the convergence inthe HNN method. The energy value, for a given iteration, iscomputed by averaging the energy values for all pairs of imagesinvolved under each experiment. We have considered each type

of images separately: (a) and (b) outdoor and indoor environ-ments, respectively, and (c) remote sensing, including syntheticimages corrupted with noise.

As one can see, the experiments E1 and E3 achieve the con-vergence with four iterations on average. The number of itera-tions required for convergence in the remainder of the experi-ments surpasses this value. This is due to the nature of the im-ages involved, where the illumination variability is affecting theconvergence, even though homomorphic filtering is applied (E2,E4, E5, E6, E7, and E8). It is worth noting that the experimentsE7 and E8, with noisy images, require the same number of iter-ations as E6 (without noise).

Table V shows the results in terms of the correct classificationfor the eight experiments. For each pair of images, we computethe PCC and Yule scores; the final result for each experiment isaveraged by the number of pairs used in such experiment. Largervalues indicate better performance. The results shown for the it-erative BRU and HNN methods are the final results obtainedwith the number of iterations given in Fig. 5, i.e., E1, E3 , E2,E5 , E4 , and E6, E7, E8 . This is the number of iter-ations fixed for the BRU approach. It is worth noting that in ourexperiments, BRU rarely achieves the convergence with such

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TABLE VAVERAGED PCC AND YULE SCORES FOR EACH METHOD AGAINST THE SET OF EXPERIMENTS

TABLE VICOMPARATIVE RESULTS IN TERMS OF EXECUTION TIME

number of iterations, i.e., this affects the final results. The MTDapproach, unlike the remaining methods, requires a sequence ofconsecutive frames, which is not available for experiments E2,E6, E7, and E8; hence it cannot be tested for such experiments.

Table VI shows the results in terms of execution time for eachmethod. All tests have been developed in MATLAB and thencompiled under MicroSoft Visual C++ 6.0 and executed on a P41.4 GHz with 512-MB RAM. The time depends on the framessize (see Table II). This size determines the number of nodesin HNN. Hence, the time is computed for this size according tothe type of environment. The time for MTD is computed for theprocessing of two consecutive frames without the labeling andclosing operations used for removing noises and shadows. Forthe iterative methods (BRU and HNN ), the results for eachexperiment must be multiplied by the number of iterations ac-cording to the results displayed in the Fig. 5. The results shownin Table VI do no include the time spent during homomorphicfiltering when applied. This time is proportional to the framesize; it has been quantified as 42.20 ms for remote sensing (400

400), 151.30 ms for indoor frames (840 760), and 381.8 msfor outdoor frames (1392 1040). One can see that the com-putational load of MAP, BRU, and HNN, which map contextualinformation, is greater than the cost required for the remainingmethods. As the times for BRU and HNN must be multipliedby the number of iterations, it is clear that they require a lot ofcomputational resources.

Figs. 6–8 show the behavior of the averaged PCC coefficientfor HNN and BRU for each experiment against the number ofiterations. The number of iterations for each experiment is thespecified above according to the results in the Fig. 5. A sim-ilar tendency is shown by the Yule coefficient. Hence, for sim-plicity, we have omitted its graphical behavior. As one can see,

HNN starts its convergence process with lower values than BRUand then HNN, after a small number of iterations, surpassesthe values of BRU. This means that the initialization processin BRU is better than the initialization in HNN. On the contrary,the HNN convergence process is faster than the convergencein BRU. This assertion can be inferred from Figs. 6–8, whereone can see that for the same number of final iterations HNNachieves better results than BRU. We have verified that BRU re-quires a greater number of iterations than HNN to achieve sim-ilar performances. This implies that BRU will increase its com-putational cost with respect to the required by HNN.

D. Summary of Results

• The best performance in terms of accuracy is obtainedwith iterative methods (HNN, BRU), where HNN achievesbetter performance than BRU for the number of iterationsused.

• Part of the improvement of HNN is due to the mapping ofthe self-information applied by HNN. This takes specialrelevance near the borders of changed areas.

• The initialization process can be important but it is not de-cisive. This conclusion was already reported in the originalwork describing the BRU method [8]. Also, this is sup-ported by the results shown in Figs. 6–8. Indeed, the resultsobtained in the first iteration are directly derived from theinitialization process. As one can see, it is clear that al-though HNN achieves worse results than BRU during thefirst iteration (i.e., initialization) they are substantially im-proved as the iteration process progresses. Hence, the ini-tialization parameter could take any value ranging be-tween 0.3 and 0.7 as two reference values without affectingthe final results.

1262 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 17, NO. 5, SEPTEMBER 2006

Fig. 6. Averaged PCC values for HNN and BRU methods against the number of iterations: Experiments E1 and E2.

Fig. 7. Averaged PCC values for HNN and BRU against the number of iterations: Experiments E3, E4 and E5.

Fig. 8. Averaged PCC values for HNN and BRU against the number of iterations: Experiments E6, E7, and E8.

• On the contrary, the time complexity is the worst for it-erative methods; perhaps its usage for critical surveillancetasks is limited. This must be improved by using parallelimplementations.

• The convergence is faster and the results better in se-quences without significant illumination changes (E1, E3)than in sequences where the illumination has changedsignificantly (E2, E4, E5). Nevertheless, one can see thatthe PCC and Yule scores in Table V for E2, E4, and E5

show acceptable values. We have verified that withouthomomorphic filtering, those scores are worse than whenthis filtering is applied. Hence, we can conclude that thehomomorphic filtering is suitable when the illuminationhas changed.

• According to the PCC and Yule scores shown in Table Vfor E7 and E8 noisy experiments, one can see that thenoise is affecting the performance as compared withthe experiment E6. Nevertheless, the robustness of HNN

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against different levels of noise is greater than the obtainedby the remainder methods. Moreover, HNN deals wellwith noisy images without the increment in the numberof iterations.

VI. CONCLUSION

In this paper, we have developed a new automatic strategyfor image change detection based on the well-founded HNNparadigm. There are strategies that exploit the performance ofspatial-contextual information. We also make use of this kindof information by integrating data and contextual informationunder the form of an energy function. The mapping of data in-formation is improved in relation to classical implementationsof this information. This improvement is achieved through theinterdata relations. Unlike classical methods, where only con-textual information is taken into account, we have also consid-ered the self-data information, also mapped under the form ofan energy function. The self-data information minimizes the er-rors derived from incorrect decisions taken by the neighborsof a pixel. This effect is especially relevant near the bordersof changed areas. The HNN approach is a suitable strategy toembed both spatial-contextual and self-data information wherean energy function is minimized by optimization. The proposedmethod has proven to be robust against noise as compared withthe remainder methods. Its accuracy performance against otherexisting strategies has been compared favorably. The proposedHNN method has been tested with additional data which are notincluded in this paper. We have verified the similar performanceas the described in this paper. The main drawback of the HNNmethod comes from its high execution time. So, for real-time re-quirements under surveillance tasks, it should be implementedunder parallel architectures.

ACKNOWLEDGMENT

The author would like to thank Prof. Dr. J. M. de la Cruz,Head of the ISCAR research group in the Universidad Com-plutense, for his support and encouragement. The author is alsograteful to the referees for their constructive criticism and sug-gestions on the original version of this paper.

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Gonzalo Pajares (M’04) received the M.Sc. andPh.D. degrees in physics from the National DistanceUniversity of Spain, Madrid, Spain, in 1987 and1995, respectively. His dissertation concerned theapplication of pattern recognition techniques tostereovision.

He joined Indra in 1990, where he worked oncritical real-time software development. He alsowas with Indra Space and INTA developing remotesensing applications. He joined the ComplutenseUniversity, Madrid, Spain, in 1995 as an Associate

Professor and, in 2004, became a Professor on the Faculty of Informatics,Department of Informatics Systems. The areas he covers are computer visualperception, artificial intelligence, and simulation. His current research interestsinclude machine visual perception, pattern recognition, and neural networks.